On the moon, the equation \( d=0.8 t^{2} \) is used to find the distance an object has fallen after \( t \) seconds. On Earth, the equation is \( d=4.9 t^{2} \). A rock is thrown from a height of 90 meters on Earth. Plot the graph of the equation on a graphing calculator or other technology to find the time taken to hit the ground to the nearest second.

To solve this problem, we start with the equation for distance on Earth, given as \( d = 4.9t^2 \). Since the rock is thrown from a height of 90 meters, we need to set the equation equal to the height from which it was thrown and solve for \( t \): \[ 90 = 4.9t^2 \] Rearranging this, we have: \[ t^2 = \frac{90}{4.9} \] Calculating this gives: \[ t^2 \approx 18.37 \] Taking the square root results in: \[ t \approx 4.29 \text{ seconds} \] Rounding to the nearest second gives \( t \approx 4 \) seconds. --- On Earth, gravity is about 9.81 m/s², which is reflected in the equation by the \( 4.9 \) factor (since \( 4.9 \) is half of \( 9.81 \)). This means the rock falls quicker than you'd think! Always remember to adjust the equation based on the height from which you're dropping or throwing the object. Opting for technology, grappling with graphing calculators or software (like Desmos) can visually illustrate where the \( t \) intercepts happen, making it easy to see the exact moment it hits the ground. This can be especially helpful when dealing with different heights or modifying conditions!